Oscillation of Fourier transforms and Markov–Bernstein inequalities

Under certain conditions on an integrable function P having a real-valued Fourier transform and such that P(0)=0, we obtain an estimate which describes the oscillation of in [-C∥P′∥∞/∥P∥∞,C∥P′∥∞/∥P∥∞], where C is an absolute constant, independent of P. Given λ>0 and an integrable function φ with a non-negative Fourier transform, this estimate allows us to construct a finite linear combination Pλ of the translates , such that with another absolute constant c>0. In particular, our construction proves the sharpness of an inequality of Mhaskar for Gaussian networks.